Effects of anisotropic supports on the stability of parametrically excited slender rotors

Nonlinear Dynamics - This study is aimed at investigating the effects of anisotropic supports on the stability of slender rotors parametrically excited by external loads. An axisymmetric shaft...     

涡轮增压器离心压气机非稳态工况流场分析

利用商用 CFD 软件对一小型车用离心压气机建立了数值模型,并将模拟结果与实验结果进行了对比:稳态的设计转速最高压比相差不超过 0.5%,最高效率相差不超过1.5%;非稳态模拟和实验得到的失速频率均为 3000Hz,模拟结果真实可信.主要利用设计转速下小流量工况时的非稳态数值模拟结果对喘振发生前离心压气机各部件的非稳态流动特点进行了详尽阐述.研究结果表明:小流量工况时离心压气机各部件均出现非稳态流动现象,这种非稳态效应在各部件中表现出不同的特点,且随着流量的减小这种非稳态效应会不断加剧.     

On forward and backward precession of rotors

Summary A study of the excitation of backward and forward precessional motion of rotors is presented. It is shown that if the rotor is supported in bearings with rotational symmetry, a rotating unbalance of the shaft cannot excite backward precessions, but if the bearings have even the slightest anisotropy, backward synchronous precessions are excited just as in the case of forward precessional motions. However, internal damping forces have a marked effect in reducing the amplitudes of the backward precessions.A rotating unbalanced non-uniform continuous circular shaft supported by anisotropic flexible bearings is studied. The governing equations of motion are derived taking into account the effects of transverse shear, gyroscopic moments, and internal and external damping.

---

Übersicht Es wird die Erregung der gleich- und gegen-läufigen Präzessionen von Rotoren untersucht. Es wird gezeigt, daß bei Rotoren in rotationssymmetrischen Lagern eine Unsymmetrie der Welle keine gegenläufigen Präzessionea verursachen kann. Bei der geringsten Anisotropie der Lagerung werden jedoch synchrone gegenläufige Präzessionen in derselben Weise wie die gleichläufigen Präzessionen erregt. Allerdings werden die gegenläufigen Präzessionen durch innere Dämpfungskräfte stärker reduziert.Es wird weiterhin eine unwuchtige Welle mit nicht-konstantem Kreisquerschnitt untersucht, die in anisotropen, nachgiebigen Lagern gelagert ist. Die Bewegungsgleichungen werden für diesen Fall unter Berücksichtigung der Querkräfte, der Kreiselmomente und der inneren und äußeren Dämpfung abgeleitet.

---

---

The author is indebted to Professor Frithiof I. Niordson for many valuable comments on the present paper.     

Dynamics and stability of elastic shaft-bearing systems with nonlinear bearing parameters

The dynamics and stability of a continuously elastic spinning shaft mounted on two dissimilar end bearings possessing nonlinear anisotropic and cross coupling stiffness and damping coefficients are analyzed. Sufficient conditions of system stability in the sense of Liapunov are derived. The developed stability criteria of the considered system have been shown to reduce to that of simpler models found in the pertinent literature. The effects of nonlinearity of bearing stiffness and damping parameters together with shaft stiffness parameter and other system parameters on the dynamic stability of the system are investigated. Several graphs demonstrating parametrically the influence of various system nondimensionalized parameters on system stability boundaries for typical cases are presented. Received on 26 October 1998     

Dynamics modeling and stability of robotic systems with discrete-time force control

This paper investigates the dynamic behavior of robotic echanical systems with discrete-time force control. Force control is associated with the constrained motion of a mechanical system. A novel approach is presented to analyze the stability and performance based on the separation of constrained and admissible motions. This results in a model representing the dynamics of the constrained motion of the system. The analysis connects the complex nonlinear model of a mechanical system to a set of abstract delayed oscillators. These oscillator models make it possible to perform a detailed closed-form mathematical analysis of the stability behavior. A planar two-degree-of-freedom (DoF) mechanism is presented as an example to illustrate the material. Results are illustrated by stability charts in the parameter space of mechanical parameters, control gains and the sampling rate.     

On self-excited whirl of rotors

Summary The paper presents a study of the non-synchronous, self-excited whirl of continuous rotors caused by internal damping forces and by hydrodynamic bearing-film forces. Taking into account gyroscopic effects, the effect of shear on bending, and external damping, the equations of motion are derived by the variational principle. By means of two mode expansions simple expressions are obtained for the limit of stability and the corresponding whirling frequency that are valid for small values of the velocity dependent forces.Numerical methods are used to obtain better approximations to the limits of stability and the whirling frequencies. The results show that for large values of internal and external damping forces and gyroscopic moments, the widely used two-mode approximation may be greatly in error. It is found that if the limit of stability is raised by adding external damping to the rotor system, then even a small amount of internal damping may considerably reduce the limit of stability. In accordance with experimental work done by other authors, it is found that for certain damping conditions a second stability region exists.

---

Übersicht Der Artikel enthält eine Untersuchung von nicht synchronen selbsterregten Kreiselbewegungen eines kontinuierlichen Rotors, die durch innere Dämpfungskräfte und hydrodynamische Schmierfilmkräfte verursacht werden. Unter Berücksichtigung des Kreiseleffektes, des Einflusses der Schubspannung auf die Biegung und der äußeren Dämpfung werden die Bewegungsgleichungen nach dem Variationsprinzip abgeleitet. Mit Hilfe einer Annäherung mit zwei Koordinatenfunktionen erhält man einfache Ausdrücke für die Stabilitätsgrenze und für die entsprechende Kreiselfrequenz, die für kleine Werte der von der Geschwindigkeit abhängigen Kräfte gelten.Um bessere Näherungen für die Stabilitätsgrenzen und die Kreiselfrequenzen zu erhalten, werden numerische Methoden angewandt. Die Ergebnisse zeigen, daß die oft benutzte Annäherung mit zwei Koordinatenfunktionen für große Werte der inneren und äußeren Dämpfung sowie bei Gegenwart von Kreiselmomenten zu großen Fehlern führen kann. Es wird gezeigt, daß schon ein kleiner Beitrag an innerer Dämpfung die Stabilitätsgrenze merkbar reduzieren kann, wenn man die Stabilitätsgrenze durch Hinzufügung äußerer Dämpfung zum Rotorsystem hinauszuschieben sucht. In Übereinstimmung mit den experimentellen Ergebnissen anderer Autoren wird gezeigt, daß ein zweiter Stabilitätsbereich für gewisse Dämpfungsbedingungen existiert.

---

---

The author wish to thank Professor Frithiof I. Niordson, lic. techn. Pauli Pedersen and lic. techn. Viggo Tvergaard for their stimulating criticism of this work.     

Turbine adapted maps for turbocharger engine matching

This paper presents a new representation of the turbine performance maps oriented for turbocharger characterization. The aim of this plot is to provide a more compact and suited form to implement in engine simulation models and to interpolate data from turbocharger test bench.The new map is based on the use of conservative parameters as turbocharger power and turbine mass flow to describe the turbine performance in all VGT positions. The curves obtained are accurately fitted with quadratic polynomials and simple interpolation techniques give reliable results.Two turbochargers characterized in an steady flow rig were used for illustrating the representation. After being implemented in a turbocharger submodel, the results obtained with the model have been compared with success against turbine performance evaluated in engine tests cells. A practical application in turbocharger matching is also provided to show how this new map can be directly employed in engine design.     

Static divergence of self-twisting composite rotors

Hydroelastic instabilities, such as static divergence and flutter, are major concerns for the safe operation of next-generation self-twisting composite marine rotors. In this paper, an efficient theoretical model was developed to obtain a first-order estimation of the static divergence speed of such rotors. The model was inspired by classical aeroelasticity theory and furnished with advanced numerical modeling. Its application to the marine environment was illustrated via a self-twisting composite propeller. The methodology is equally applicable to other structures, such as tidal and wind turbines.     

Analysis of erosion behaviour in a turbocharger radial turbine

An analysis of the erosion behaviour of a turbocharger radial turbine is presented. The solution domain includes both sides of the radial turbine scroll with double intake and the rotor channel. In the analysis a dilute gas-particle flow assumption is employed. The gas turbulence is defined by the k-ε model. In solving the gas phase equation, the computer code Harwell-FLOW3D is employed, which is based on a finite volume formulation using non-orthogonal body-fitted structured gridding and a pressure correction method. The particle phase is described by a Lagrangian approach, while particle paths are computed deterministically, neglecting the turbulent dispersion. For the computation of particle trajectories the code PTRACK is employed, which has been developed at ABB. Computations are carried out for several particle size classes. The results show that particles are thrown back into the scroll by the rotor at high rates. This seems to be the main source of erosion effects in the scroll. It has been observed that particles are unequally distributed between the scroll sides on their re-entry, resulting in greater erosion on one of the scroll sides. The maximum erosion along the scroll is found to be likely to occur near the scroll end.     

Flow visualization and analysis of aerodynamically detuned supersonic rotors

The effect of aerodynamic detuning on the supersonic steady and unsteady blade passage flow field is experimentally investigated on a free surface water table by means of color Schlieren and shadowgraph flow visualization techniques. Two aerodynamic detuning mechanisms are considered: (1) alternate circumferential spacing of adjacent airfoils; (2) the replacement of alternate airfoils with splitters. The steady flow visualization demonstrates the significant effect of aerodynamic detuning on the passage flow field and, in particular, the shock wave-airfoil surface intersection locations. The unsteady flow visualization studies show the importance of the interblade phase angle. A mathematical model is also described and utilized to demonstrate the enhanced aeroelastic stability associated with the altered cascade passage shock wave structure due to these aerodynamic detuning mechanisms.List of symbols a dimensionless perturbation sonic velocity - C airfoil chord - I _x mass moment of inertia - k reduced frequency, k = C/2 - K spring constant - M _R dimensionless unsteady aerodynamic moment - M Mach number - P _split split splitter circumferential spacing - P _start splitter leading edge location - R reference full chord airfoil - R _s reference splitter - Sp _chord splitter chord length - u dimensionless perturbation chordwise velocity - v dimensionless perturbation normal velocity - amplitude of oscillation - interblade phase angle - level of aerodynamic detuning - undamped natural torsional frequency - ₀ reference frequency - flutter frequency     

Basic design scheme for wave rotors

Pressure wave devices use shock waves to transfer energy directly between fluids without additional mechanical components, thus having the potential for increased efficiency. The wave rotor is a promising technology which uses shock waves in a self-cooled dynamic pressure exchange between fluids. For high-pressure, high-temperature topping cycles, it results in increased engine overall pressure and temperature ratio, which in turn generates higher efficiency and lower specific fuel consumption. Designing a wave rotor mainly focuses on predicting the behavior of shock and expansion waves. The extant literature presents numerous examples of wave rotor designs, but most of them rely on complicated numerical analyses as well as computer code developed specifically for this application. This paper presents an initial scheme used for designing wave rotors employing thermodynamic and gasdynamic analysis as well as computational fluid dynamic analysis. Basic theory and a simplified model of the wave rotor are used to predict the travel time and strength of waves. The model is then refined using a more advanced numerical scheme on the basis of the Lax–Wendroff method and FLUENT, a commercial CFD code.

---

Research was conducted while F. Iancu was a Ph.D. candidate at Michigan State University.     

Automatic two-plane balancing for rigid rotors

We present an analysis of a two-plane automatic balancing device for rigid rotors. Ball bearings, which are free to travel around a race, are used to eliminate imbalance due to shaft eccentricity or misalignment. The rotating frame is used to derive autonomous equations of motion and the symmetry breaking bifurcations of this system are investigated. Stability diagrams in various parameter planes show the coexistence of a stable balanced state with other less desirable dynamics.     

Parallel solution of lifting rotors in hover and forward flight

An implicit unsteady, multiblock, multigrid, upwind solver including mesh deformation capability, and structured multiblock grid generator, are presented and applied to lifting rotors in both hover and forward flight. To allow the use of very fine meshes and, hence, better representation of the flow physics, a parallel version of the code has been developed. It is demonstrated that once the grid density is sufficient to capture enough turns of the tip vortices, hover exhibits oscillatory behaviour of the wake, even using a steady formulation. An unsteady simulation is then presented, and detailed analysis of the time‐accurate wake history is performed and compared to theoretical predictions. Forward flight simulations are also presented and, again, grid density effects on the wake formation investigated. Parallel performance of the code using up to 1024 CPU's is also presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Vibrations of asymmetric rotors supported by asymmetric bearings

Summary In this paper the vibrations of an asymmetric flexible rotor supported by asymmetric bearings is theoretically analyzed by using Galerkin's method and the perturbation method, and numerically calculated. The effects of the asymmetries of the rotor and the bearings and the changes of the main instability regions and the ultraharmonic resonance due to the external and internal dampings are investigated. The experimental tests are performed on a smaller laboratory model in order to verify the validity of the theoretical results.The following results are obtained; the instability regions divide into two ones by the lack of the symmetries in the rotor and the bearings. The ultraharmonic resonances appear at fractional values of the main critical speed. The character of the internal damping is changed by the magnitude of the asymmetries of the rotor and the bearings.

---

Übersicht Es werden die Schwingungen eines unsymmetrischen Rotors in unsymmetrischen Lagern analysiert. Dabei wird sowohl das Galerkin-Verfahren als auch eine Störungs-Methode angewendet. Untersucht werden die durch die Unsymmetrien von Rotor und Lagerung entstehenden Effekte sowie die durch äußere und innere Dämpfung entstehenden Änderungen der wichtigsten Instabilitätsbereiche und der ultraharmonischen Resonanzen. Zur Stützung der theoretischen Ergebnisse wurden Versuche an einem Laboratoriumsmodell durchgeführt.Die wichtigsten Ergebnisse sind folgende: Die Instabilitätsbereiche werden durch die Unsymmetrie von Rotor und Lagerung aufgespalten; ultraharmonische Resonanzen treten bei gebroochenen Werten der Hauptkritischen auf; der Charakter der inneren Dämpfung wird durch die Größe der Unsymmetrien von Rotor und Lagerung verändert.

     

Conditions for noncircular whirling of nonlinear isotropic rotors

Nonlinear rotors are often considered as potential sources of chaotic vibrations. The aim of the present paper is that of studying in detail the behaviour of a nonlinear isotropic Jeffcott rotor, representing the simplest nonlinear rotor. The restoring and damping forces have been expanded in Taylor series obtaining a Duffing-type equation. The isotropic nature of the system allows circular whirling to be a solution at all rotational speeds. However there are ranges of rotational speed in which this solution is unstable and other, more complicated, solutions exist.The conditions for stability of circular whirling are first studied from closed form solutions of the mathematical model and then the conditions for the existence of solutions of other type are studied by numerical experimentation. Although attractors of the limit cycle type are often found, chaotic attractors were identified only in few very particular cases. An attractor supposedly of the last type reported in the literature was found, after a detailed analysis, to be related to a nonchaotic polyharmonic solution.As the typical unbalance response of isotropic nonlinear rotors has been shown to be a synchronous circular whirling motion, the convergence characteristics of Newton-Raphson algorithm applied to the solution of the set of nonlinear algebraic equations obtained from the differential equations of motion are studied in some detail. c damping coefficient i imaginaty unit (i=% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbqfgBHr% xAU9gimLMBVrxEWvgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvA% Tv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9% vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea% 0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabe% aadaabauaaaOqaamaakaaabaGaeyOeI0IaaGymaaWcbeaaaaa!3E66!\[\sqrt { - 1}\]) k stiffness m mass t time x _istate variables i=1, 4 z complex co-ordinate (z=x+iy)[J] Jacobian matrix Oxyz inertial co-ordinate frame Oz rotating co-ordinate frame perturbation term eccentricity complex co-ordinate (=+i) system eigenvalues nonlinearity parameter nondimensional time phase spin speed u nonrotating t rotating0 amplitude t nondimensional termsNomenclature